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Stationary solutions of the Maxwell-Dirac and the Klein-Gordon-Dirac equations. (English) Zbl 0869.35105

Interaction of an electron with its own electromagnetic field is described by the Maxwell-Dirac (M-D) equations. They are written as: \[ (i\gamma^\mu\partial_\mu- i\gamma^\mu A_\mu)\psi- m\psi= 0,\quad \partial_\mu A^\mu=0,\quad 4\pi\partial_\mu\partial^\mu A^\nu= J^\nu\quad\text{in }\mathbb{R}\times\mathbb{R}^3, \] where \(\gamma^\mu=\left[\begin{matrix} 0 &\sigma^\mu\\ -\sigma^\mu & 0\end{matrix}\right]\) for \(\mu=1,2,3\), and \(\gamma^0=\left[\begin{matrix} I & 0\\ 0 &-I\end{matrix}\right]\). Here \(\sigma^\mu\) are the Pauli matrices, \(I\) is the identity matrix. An extensive literature exists concerning special cases of the M-D equations. For example, Georgiev proved existence of global solutions for a certain class of initial values. Using approximations to solutions of M-D equations Wakano proved in 1996 the existence of stationary, i.e. soliton-like standing waves.
The present authors consider exact equations (not approximations) and apply a variational principle of Esteban and Séré to the energy of the M-D system. The solution is the critical point of the energy functional. A similar argument works also for the Klein-Gordon-Dirac equations, again showing existence of soliton-like solutions. These results are important, since as is well-known soliton waves have many properties of both waves and particles.
Reviewer: V.Komkov (Roswell)

MSC:

35Q75 PDEs in connection with relativity and gravitational theory
83C15 Exact solutions to problems in general relativity and gravitational theory
49S05 Variational principles of physics
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